A geometric transition from hyperbolic to anti-de Sitter geometry

Danciger, Jeffrey

doi:10.2140/gt.2013.17.3077

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A geometric transition from hyperbolic to anti-de Sitter geometry

Jeffrey Danciger

Geometry & Topology 17 (2013) 3077–3134

DOI: 10.2140/gt.2013.17.3077

Abstract

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti-de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the “other side” of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these methods in the case when the manifold is the unit tangent bundle of the $(2, m, m)$ triangle orbifold for $m \geq 5$ .

Keywords

geometric transition, hyperbolic, AdS, cone manifold, tachyon, projective structure, transitional geometry, half-pipe geometry

Mathematical Subject Classification 2010

Primary: 57M50

Secondary: 53C15, 53B30, 20H10, 53C30

References

Publication

Received: 25 February 2013

Accepted: 26 June 2013

Published: 17 October 2013

Proposed: Danny Calegari

Seconded: Benson Farb, Jean-Pierre Otal

Authors

Jeffrey Danciger
Department of Mathematics University of Texas – Austin 1 University Station C1200 Austin, TX 78712-0257 USA
http://www.ma.utexas.edu/users/jdanciger/

Volume 17, issue 5 (2013)